Constraint Satisfaction taking advantage of internal structure of states when paths don’t matter.

Constraint Satisfaction

taking advantage of internal structure of stateswhen paths don’t matter

D Goforth - COSC 4117, fall 2003 2

Constraint satisfaction problems (CSPs)

no prescribed start state path does not matter acceptable states must satisfy

constraints functions of state variables

among acceptable states, goal states will optimize an objective function


Example problem – timetable of university (1)

state - collection of schedule objects schedule

professor(~102) (~350 at LU) course (~103) time slot (~10) (16 at LU) location (~102)

constraints objective function



constraints professors assigned to courses one course per location per time slot location capacity >= course (projected)

enrollment professors assigned only once per time

slot rooms assigned only once per time slot courses in program in different time slots



objective function minimize total course enrollment in early

morning time slots distribute program courses across days concentrate professor assignments on

same day accommodate special requests


Solving CSPs

analysis of problem state structure is critical

two basic approaches: start state is null: assign state variables

one at a time and backtrack if constraint is violated

assign variables ‘randomly/intelligently’ to make a start state and use greedy search by changing variable assignments



analyzing the problem e.g. fix time slots for all weekly sessions

of a course (tutorials?) problem solving approaches

allocate courses to rooms and times one after the other

assign all courses then start moving those involved in constraint violations


CSP – problem definition

state representation (R&N) state variables X1, X2, …, Xn

Xi from domain Di

constraints C1, C2, …, Cm

Cj are functions of Xi (unary, binary, … fns) solution – assignment of values vi to all Xi so

all constraint functions Cj are satisfied objective function – optimized function of Xi

over solution set



variables: array of assignments of courses (ck) to locations (loci)and time (tj) slots ck:prof,enroll,prog loci: capacity tj: time, day A[loc,t] = ck

constraints: loc.capacity >= c.enroll (unary) if (loci1 != loci2) A[loci1,tj].prof != A[loci2,tj].prof


Modelling a state as constraint graph

variables are nodes

loc 1: 25 loc 2: 100

t 1

t 2

t 3

c 1: Ann 20

c 2: Ben 45

c 3: Cec 15

c 4: Ben 80

c 5: Ann 90



unary constraints reduce domains

loc 1: 25 loc 2: 100

t 1

t 2

t 3

c 1: Ann 20

c 2: Ben 45

c 3: Cec 15

c 4: Ben 80

c 5: Ann 90

for A(loc1, t) {c1,c3}

for A(loc2, t) {c1,c2,c3,c4,c5}



binary constraints are edges

loc 1: 25 loc 2: 100

t 1

t 2

t 3

c 1: Ann 20

c 2: Ben 45

c 3: Cec 15

c 4: Ben 80

c 5: Ann 90



A[loc1,tj].prof != A[loc2,tj].prof



multiple constraints are ‘hubs’

loc 1: 25 loc 2: 100

t 1

t 2

t 3

c 1: Ann 20

c 2: Ben 45

c 3: Cec 15

c 4: Ben 80

c 5: Ann 90



each course in one slot only


Variable domains

discrete finite discrete infinite continuous

Domains may be distinct or shared among variables (typical)


Constraints

enumerations of possible combinations

functions of variables linear, non-linear, … unary, binary,…


Solution by incremental formulation

initial state – no variable assigned successor function for transition:

pick a variable and set a value that does not conflict with previous assignments by applying constraint functions

goal – all variables assigned


Solution by incremental formulation

advantage depth of search is limited to number of

variables in state depth limited dfs is natural algorithm

‘backtracking search’

Backtracking searchloc 1: 25

loc 2: 100

c 1: Ann 20

c 2: Ben 45

c 3: Cec 15

c 4: Ben 80

c 5: Ann 90

c 5

c 1 c 5

c 5

c 5

c 1

c 5

c 5


Efficient Backtracking search (1)

Forward checking - Propagate constraints when variable is assigned, use

constraints to reduce value sets of remaining variables

e.g. when course is assigned: remove course from domain of all

remaining variables remove courses with same prof from

allocations in same time slot



Variable ordering – which variable to assign a value next? MRV heuristic – pick most constrained

variable (minimum remaining values) e.g. – book small room first

degree heuristic – pick variable involved in most constraints (propagates and prunes most)



Value ordering: once variable is selected, which value to try first? least constraining value heuristic – pick

value that reduces value sets of remaining variables the least

e.g. when assigning a course: pick location with smallest capacity that will

accommodate the course (risk?)



Arc consistency: stronger constraint propagation check for consistency between variables

after value is set

forward checking

remove variables

arc consistency

?



k-consistency: extending arc-consistency

forward checking

remove variables

arc consistency


Norvig and Russell example

chapter04b.ps


DFS ‘chronological’ backtracking

when search fails X6(no value can be set for variable)

back up one level, try again v5 v5

useless IFno constraint between X5, X6

X1

X2

X3

X4

X5

X6


DFS backjumping

when search fails X6(no value can be set for variable)

look at conflict set of X6 {X3, X1}

jump back to change v3 v3

X1

X2

X3

X4

X5

X6


Solution by local search start state has values for every

variable state variables X1=v1, X2=v2, …, Xn=vn

constraints C1, C2, …, Cm some satisfied, some violated

objective function not optimized successor function changes one or

more variables – evaluate to minimize conflicts


Solution by local search

timetable example: start state – assign all courses to

room/time slot change assignments to reduce

conflicts


advantages of local search solves some problems very fast flexible in online situations – revised

conditions can be re-solved with minimal changes e.g. course enrollment projections turn

out to be wrong – conflicts with room sizes

takes advantage of any known partial solutions installed in start state


Constraint graph structure independent variables (unary constraints

only) easy but rare tree graphs – process variables in top-down

order from any node as root general graphs – may be reducible (by

removing and assigning some nodes) to trees

general graphs – may be reducible by clustering into subgraphs


Examples

crossword puzzle construction p.158 #5.4

cryptarithmetic puzzle – section 5.2 other example

Sudoku puzzle

Sudoku constraintsFrom rules

Constraints on all cells:

For rows, columns, squares

Every integer occurs

No integer is repeated

From data (particular game)

Constraints on some cells

Specific value

Underconstrained – many solutions

Overconstrained – no solutions

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Constraint Satisfaction taking advantage of internal structure of states when paths don’t matter.

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